On a Riemann-Roch formula for stacks with finite cyclotomic inertia

TL;DR

This paper extends Toen's Riemann-Roch map to stacks with finite cyclotomic inertia, providing a geometric decomposition and explicit inverse in certain cases, advancing the understanding of K-theory for algebraic stacks.

Contribution

It offers a geometric interpretation of Vezzosi-Vistoli decomposition and defines a Riemann-Roch map for stacks with finite cyclotomic inertia, including explicit inverse computations.

Findings

The Riemann-Roch map is an isomorphism under finiteness conditions.

Explicit inverse maps are computed in favorable cases.

The approach recovers Toen's map for tame Deligne-Mumford stacks.

Abstract

B. Toen defined a Riemann-Roch map from the rational algebraic K-theory of a tame Deligne-Mumford quotient stack to the \'etale K-theory of its inertia. He proved that this map is an isomorphism and that it is covariant with respect to proper maps. Moreover G. Vezzosi and A. Vistoli proved a decomposition theorem for the equivariant K-theory of a noetherian scheme. In this paper we give a geometric definition of the Vezzosi-Vistoli decomposition, interpreting the pieces as corresponding to the components of the cyclotomic inertia. When the map from the cyclotomic inertia to the stack is finite, we can define a Riemann-Roch map in Toen's style. We prove that this map is an isomorphism and it is covariant with respect to proper relatively tame maps; moreover in some favourable circumstances we explicitly compute its inverse map, and show that we can recover Toen's one when the stack is tame Deligne-Mumford.